Table Of ContentGREEN’S -ORDER AND
J
THE RANK OF TROPICAL MATRICES
MARIANNE JOHNSON1 and MARK KAMBITES2
School of Mathematics, University of Manchester,
Manchester M13 9PL, England.
2
1
0 Abstract. WestudyGreen’sJ-orderandJ-equivalenceforthesemi-
2 groupofalln×nmatricesoverthetropicalsemiring. Wegiveanexact
n characterisation oftheJ-order, intermsof morphismsbetween certain
a tropical convex sets. We establish connections between the J-order,
J
isometries of tropical convex sets, and various notions of rank for trop-
4 ical matrices. We also study the relationship between the relations J
and D; Izhakian and Margolis have observed that D6=J for the semi-
]
group of all 3×3 matrices over the tropical semiring with −∞, but in
A
contrast, we show that, D = J for all full matrix semigroups over the
R
finitary tropical semiring.
.
h
t 1. Introduction
a
m
Tropical algebra (also known as max-plus algebra or max algebra) is the
[
algebra of the real numbers (typically augmented with , and sometimes
−∞
2 also with + ) under the operations of addition and maximum. It has been
v ∞
an active area of study in its own right since the 1970’s [14] and also has
7
applications in diverse areas such as analysis of discrete event systems [22],
0
7 combinatorial optimisation and scheduling problems [8], formal languages
2 and automata [34, 37], control theory [12], phylogenetics [19], statistical
.
2 inference [33], biology [7], algebraic geometry [4, 32, 35] and combinato-
0
rial/geometric group theory [5]. Tropical algebra and many of its basic
1
properties have been independently rediscovered many times by researchers
1
: in these fields.
v
Many problems arising from these application areas are naturally ex-
i
X
pressed using (max-plus) linear equations, so much of tropical algebra con-
r cernsmatrices. Froman algebraicperspective, akey objectisthesemigroup
a
of all square matrices of a given size over the tropical semiring. This semi-
group obviously plays a role analogous to that of the full matrix semigroup
over a field. Perhaps less obviously, the relative scarcity of invertible ma-
trices over an idempotent semifield means that much less is to be learnt
here by studying only invertible matrices. Many classical matrix problems
can be reduced to questions about invertible matrices, and hence about the
general linear group. In tropical algebra, however, there is typically no such
reduction, so problems often entail a detailed analysis of non-invertible ma-
trices. In this respect, the full matrix semigroup takes on the mantle of the
2000 Mathematics Subject Classification. 20M10; 14T05, 52B20.
1Email [email protected].
2Email [email protected].
1
2 GREEN’S J-ORDER AND THE RANK OF TROPICAL MATRICES
general linear group. Its algebraic structure has thus been the subject of
considerable study, at first on an ad hoc basis, but more recently moving
towards asystematic understandingusingthetools ofsemigrouptheory (see
for example [16, 23, 25, 27, 29, 37]).
Green’s relations [21, 10] are five equivalence relations ( , , , and
L R H D
) and three pre-orders ( , and ) which can be defined upon any
L R J
J ≤ ≤ ≤
semigroup, and which encapsulate the structure of its maximal subgroups
and principal left, right and two-sided ideals. They are powerful tools for
understandingsemigroups and monoids, and play a key role in almost every
aspect of modern semigroup theory. The relations , , , and can
L R
≤ ≤ L R H
bedescribedin generality for thefullmatrix semigroup over a semiringwith
identity, and hence present no particular challenge in the tropical case; see
[29] or Section 3 below for details. In [29], we initiated the study of Green’s
relationsintropicalmatrixsemigroups,bydescribingtheremainingrelations
in the case of the 2 2 tropical matrix semigroup. In [23], Hollings and the
×
second author gave a complete description of the -relation in arbitrary
D
finite dimensions, based on some deep connections with the phenomenon of
duality between the row and column space of a tropical matrix.
In the present paper, we turn our attention to the equivalence relation
and pre-order in the full tropical matrix semigroup of arbitrary di-
J
J ≤
mension. In the classical case of finite-dimensional matrices over a field, it
is well known that the relations and coincide. Previous work of the
D J
authors showed that this correspondence holds for 2 2 tropical matrices
×
[29], but Izhakian and Margolis [26] have shown that it does not extend
to higher dimensional tropical matrix semigroups over the tropical semiring
with ; indeed, they have found an example of a -class in the 3 3
−∞ J ×
tropical matrix semigroup which contains infinitely many -classes.
D
Our main technical result (Theorem 5.3) gives a precise characterisation
of the -order (and hence also of -equivalence) in terms of morphisms
J J
between certain tropical convex sets: specifically, A B exactly if there
J
≤
exists aconvex setY such thattherow spaceof B mapssurjectively ontoY,
and the row space of A embeds injectively into Y. Using duality theorems,
we also show (Theorem 5.4) that if A B then the row space of A admits
J
≤
an isometric (with respect to the Hilbert projective metric) embedding into
the row space of B, but we show that in general it need not admit a linear
embedding. From these results, we are able to deduce that the semigroup of
n n matrices over the finitary tropical semiring (without ) does satisfy
× −∞
= .
D J
In the classical case of a finite-dimensional full matrix semigroup over a
field, it is well known that the -relation (and the -relation, with which it
J D
coincides) encapsulates the concept of rank, with the -order corresponding
J
to the obvious order on ranks. Thus, for the semigroup of matrices over a
more general ring or semiring, the -class of a matrix may be thought of
J
as a natural analogue of its rank. This idea is rather different to traditional
notions of rank since it is non-numerical, taking values in a poset (the -
J
order), rather than the natural numbers.
For tropical matrices, several different (numerical) notions of rank have
been proposed and studied, both separately and in relation to one another
GREEN’S J-ORDER AND THE RANK OF TROPICAL MATRICES 3
(see for example [1, 2, 9, 17, 28, 36]). Each of these clearly has merit for
particular applications, but overall we suggest that the proliferation of in-
compatible definitions is evidence that the kind of information given by the
“rank”ofaclassicalmatrixcannot,inthetropicalcase, beencapsulatedina
single naturalnumber. We believe that the -class of a matrix may serve as
J
a “general purpose” analogue of rank for tropical mathematics, and partly
with this in mind, the final section of this paper discusses the relationship
between -class and some existing notions of rank.
J
2. Preliminaries
In this section we briefly recall the foundational definitions of tropical
algebra, and establish some elementary properties which will be required
later.
The finitary tropical semiring FT is the semiring (without additive iden-
tity) consisting of the real numbers under the operations of addition and
maximum. We write a b to denote the maximum of a and b, and a b
⊕ ⊗
or just ab to denote the sum of a and b. Note that both operations are
associative and commutative and that distributes over .
The tropical semiring T is the finitar⊗y tropical semiring⊕augmented with
an extra element which acts as a zero for addition and an identity for
maximum. The co−m∞pleted tropical semiring T is the tropical semiring aug-
mented with an extra element + , which acts as a zero for both maximum
∞
and addition, save that
( )(+ ) = (+ )( ) = .
−∞ ∞ ∞ −∞ −∞
Thus FT T T and we call the elements of FT finite elements.
⊆ ⊆
For any commutative semiring S, we denote by M (S) the set of all n n
n
×
matrices with entries drawn from S. This has the structure of a semigroup,
under the multiplication induced from the semiring operations in the usual
way.
We extend the usual order on R to a total order on T and T by setting
< x < + for all x R≤. Note that a b = a exactly if b a. The
−se∞mirings FT ∞and T admi∈t a natural order-⊕reversing involution ≤x x,
7→ −
where of course ( )= + and (+ )= .
− −∞ ∞ − ∞ −∞
For S FT,T,T we shallbeinterested inthespace Sn of affine tropical
∈{ }
vectors. We write x for the ith component of a vector x Sn. We extend
i
∈
and to Sn componentwise so that (x y) =x y and x y exactly
i i i
⊕ ≤ ⊕ ⊕ ≤
if x y for all i. We define a scaling action of S on Sn by
i i
≤
λ (x ,...,x ) = (λ x ,...,λ x )
1 n 1 n
⊗ ⊗ ⊗
for each λ S and each x Sn. Similarly, for S FT,T we extend the
∈ ∈ ∈ { }
involution x x on S to Sn by defining ( x) = (x ). The scaling and
i i
7→ − − −
operations give Sn the structure of an S-module (sometimes called an
⊕
S-semimodule since the operation does not admit inverses).
⊕
From affine tropical n-space we obtain projective tropical (n 1)-space
−
(denoted PFT(n−1), PT(n−1) or PT(n−1) as appropriate) by identifying two
vectors if one is a tropical multiple of the other by an element of FT.
4 GREEN’S J-ORDER AND THE RANK OF TROPICAL MATRICES
An S-linear convex set in Sn is a subset closed under and scaling by
⊕
elements of S, that is, an S-submodule of Sn. If B Sn then the (S-linear)
⊆
convex hull of B is smallest convex set containing B, that is, the set of all
vectors in Sn which can bewritten as tropical linear combinations of finitely
many vectors from B. Given two convex sets X Sn and Y Sm, we say
⊆ ⊆
that f : X Y is a linear map from X to Y if f(x x′) = f(x) f(x′)
→ ⊕ ⊕
and f(λ x) = λ f(x) for all x,x′ X and all λ S.
⊗ ⊗ ∈ ∈
SinceeachconvexsetX Sn isclosedunderscaling,itinducesasubsetof
⊆
thecorrespondingprojective space, termedtheprojectivisation of X. Notice
that one convex set contains another exactly if there is a corresponding
containment of their projectivisations.
Given a matrix A M (S) we definethe row space of A, denoted R (A),
n S
∈
tobetheS-linearconvexhulloftherowsofA. ThusR (A) Sn. Similarly,
S
⊆
wedefinethecolumn space C (A) Sn to betheS-linear convex hullofthe
S
⊆
columns of A. We shall also be interested in the projectivisation of C (A),
S
which we call theprojective column space of A and denotePC (A). Dually,
S
the projective row space PR (A) is the projectivisation of the row space of
S
A.
We define a scalar product operation Tn Tn Tn on affine tropical
× →
n-space by setting
x y = max λ T : λ x y .
h | i { ∈ ⊗ ≤ }
This is a residual operation in the sense of residuation theory [6], and has
been frequently employed in max-plus algebra. Notice that x y = +
h | i ∞
if and only if for each i either x = or y = + . Thus x x = +
i i
if and only if x ,+ for all−i∞. It also follo∞ws that ifhx,|y i T wi∞th
i
x = ( ,..., ∈){−th∞en x∞}y T. Similarly, we note that x y∈ =
6 −∞ −∞ h | i ∈ h | i −∞
ifandonlyifthereexistsj suchthateitherx = + = y ory = = x .
j j j j
Thus if x,y FTn then x y FT. ∞ 6 −∞ 6
∈ h | i ∈
Lemma 2.1. Let x,y Tn with x = y. If x y = + then y x = .
∈ 6 h | i ∞ h | i −∞
Proof. Since x = y there exists j such that x = y . Now, x y = +
j j
6 6 h | i ∞
implies thatforeach ieither x = or y = + . Thuseitherx = =
i i j
−∞ ∞ −∞ 6
y or y = + = x and hence, by the remarks preceding the lemma, we
j j j
find that y ∞x 6= . (cid:3)
h | i −∞
We define a distance function on Tn by d (x,y) = 0 if x is a finite scalar
H
multiple of y and
d (x,y) = ( x y y x )
H
− h | i⊗h | i
otherwise. By Lemma 2.1, it is easy to see that d (x,y) = for all
H
x,y Tn. Thus d (x,y) = + unless both x y and y 6 x−a∞re finite.
H
More∈over, if x y , y x F∞T then it is easyh to| chieck thhat|di(x,y) 0.
H
h | i h | i ∈ ≥
It is also easily verified that d is invariant under scaling x or y by finite
H
scalars and hence is well-defined on PT(n−1), PT(n−1) and PFT(n−1). For
x,y FTn we see that d (x,y) FT. In fact, it can be shown that d
H H
∈ ∈
is a metric on PFT(n−1) and an extended metric on PT(n−1) and PT(n−1),
called the (tropical) Hilbert projective metric (see [23, Proposition 1.6], for
example). In particular, d induces obvious definitions of isometry and
H
isometric embeddings between subsets of tropical projective spaces.
GREEN’S J-ORDER AND THE RANK OF TROPICAL MATRICES 5
NowletS FT,T andletA M (S). Following [13]and[18]wedefine
n
∈ { } ∈
amapθ :R (A) C (A)byθ (x) = A ( x)T forallx R (A). Dually,
A S S A S
→ ⊗ − ∈
we define θ′ : C (A) R (A) by θ′ (x) = ( x)T A for all x C (A).
A S → S A − ⊗ ∈ S
We call θ and θ′ the duality maps for A. Notice that the duality maps
A A
do not make sense over S = T, as the involution x x is not defined for
7→ −
x = . The following lemma recalls some known properties of the duality
−∞
maps which we shall need.
Lemma 2.2. (Properties of the duality maps [13, 18, 23]).)
Let S FT,T and let A M (S).
n
∈ { } ∈
(i) θ andθ′ are mutuallyinversebijectionsbetweenR (A)andC (A).
A A S S
(ii) For all x,y R (A), x y if and only θ (y) θ (x).
S A A
∈ ≤ ≤
For all x,y C (A), x y if and only θ′ (y) θ′ (x).
∈ S ≤ A ≤ A
We say that θ and θ′ are order reversing.
A A
(iii) For all x R (A) and all λ FT, θ (λ x) = λ θ (x).
S A A
For all x ∈ C (A) and all λ ∈FT, θ′ (λ⊗x) = −λ⊗θ′ (x).
∈ S ∈ A ⊗ − ⊗ A
We say that θ and θ′ preserve scaling by finite scalars.
A A
Of the properties in the lemma, part (i) is established for S = FT in [18]
and for S = T in [13]. Part (ii) is shown in [13]. Part (iii) is proved in [23],
which also includes an expository account of the other two parts.
We now recall the “metric duality theorem” of [23].
Theorem 2.3. (Metric duality theorem.)
Let S FT,T and let A M (S). Then the duality maps θ and
n A
∈ { } ∈
θ′ induce mutually inverse isometries (with respect to the Hilbert projective
A
metric) between PR (A) and PC (A).
S S
3. Green’s Relations
Green’srelationsarefiveequivalencerelationsandthreepre-orders,which
canbedefinedonanysemigroup,andwhichtogetherdescribethe(left, right
andtwo-sided) principalidealstructureof thesemigroup. We giveherebrief
definitions; for fuller discussion, proof of claimed properties and equivalent
formulations, we refer the reader to an introductory text such as [24].
Let S be any semigroup. If S is a monoid, we set S1 = S, and otherwise
we denote by S1 the monoid obtained by adjoining a new identity element
1 to S. We define a binary relation on S by a b if aS1 bS1, that
R R
≤ ≤ ⊆
is, if either a = b or there exists q with a = bq. We define another relation
by a b if and only if aS1 = bS1.
R R
The relations and are the left-right duals of and , so a b if
L R L
≤ L ≤ R ≤
S1a S1b, and a b if S1a = S1b. The relations and are two-sided
J
⊆ L ≤ J
analogues, so a b if S1aS1 S1bS1, and a b if S1aS1 = S1bS1. We
J
≤ ⊆ J
also define a relation by a b if a b and a b. Finally, the relation
H H L R
is defined by a b if there exists an element c S such that a c and
D D ∈ R
c a.
L
The relations , , , and are equivalence relations; this is trivial
R L H J D
in the first four cases, but requires slightly more work in the case of .
D
The relations , and are pre-orders (reflexive, transitive binary
R L J
≤ ≤ ≤
relations) each of which induces a partial order on the equivalence classes of
the corresponding equivalence relation.
6 GREEN’S J-ORDER AND THE RANK OF TROPICAL MATRICES
The study of Green’s relations for the full tropical matrix semigroups
was begun (in the 2 2 case) by the authors [29] and continued in greater
×
generality by Hollings and the second author [23]. Some key results of those
papers are summarised in the following two theorems; see [23, Proposition
3.1], [23, Theorem 5.1], [23, Theorem 5.5] and [23, Theorem 3.5] for full
details and proofs.
Theorem 3.1. (Known characterisations of Green’s Relations.)
Let A,B M (S) for S FT,T,T .
n
∈ ∈ { }
(i) A B if and only if R (A) R (B);
L S S
≤ ⊆
(ii) A B if and only if R (A) = R (B);
S S
L
(iii) A B if and only if C (A) C (B);
R S S
≤ ⊆
(iv) A B if and only if C (A) = C (B);
S S
R
(v) A B if and only if R (A) = R (B) and C (A) = C (B);
S S S S
H
(vi) A B ifandonly ifC (A)andC (B)are isomorphic asS-modules;
S S
D
(vii) A B ifandonlyifR (A)andR (B)areisomorphic asS-modules.
S S
D
Theorem 3.2. (Inheritance of , , and .)
L R H D
Consider M (FT) M (T) M (T). Each of Green’s pre-orders ,
n n n L
and equivalence ⊆relations ⊆, , and in M (FT) or M (T) is≤the
R n n
≤ L R H D
restriction of the corresponding relation in M (T).
n
The -relation and pre-order for the semigroups M (S) with S
J n
J ≤ ∈
FT,T,T have so far remained rather mysterious. We shall give a charac-
{ }
terisation of the -relation in these tropical matrix semigroups.
J
4. The -relation is inherited
J
Inmanyapplications onewishestoworkwiththetropicalsemiringT,but
for theoretical purposes it is often nicer to work over the finitary tropical
semiringFTor the completed tropical semiringT. Thefollowing resultis an
analogue for the -order and -equivalence of Theorem 3.2 above, saying
that these relationJs in a full mJatrix semigroup over T and FT are inherited
from the corresponding semigroup over T. Hence, in order to understand
these relations in all three cases, it suffices to study them for T, and weshall
for much of the remainder of the paper work chiefly with T.
Proposition 4.1. (Inheritance of .)
J
Consider M (FT) M (T) M (T).
n n n
⊆ ⊆
(i) Let A,B M (FT). Then
n
∈
A B in M (FT) if and only if A B in M (T).
J n J n
≤ ≤
(ii) Let A,B M (T). Then
n
∈
A B in M (T) if and only if A B in M (T).
J n J n
≤ ≤
Proof. (i) It is clear that if A B in M (FT) then A B in M (T).
J n J n
Suppose now that A B in M≤ (T). Thus there exist P,≤Q M (T) such
J n n
≤ ∈
that A= PBQ giving
n n
A = P B Q (4.1)
i,j i,k k,l l,j
Mk=1Ml=1
GREEN’S J-ORDER AND THE RANK OF TROPICAL MATRICES 7
for all i,j 1,...,n . Since P,B,Q each have finitely many entries we
may choose∈δ{ R such}that:
∈
(1) δ p+b+q b′ p′ for every pair of (necessarily finite) entries b,b′
≤ − −
in B and all finite entries p,p′ in P and q in Q;
(2) δ p+b+q b′ q′ for every pair of (necessarily finite) entries b,b′
≤ − −
in B and all finite entries p in P and q,q′ in Q;
(3) 2δ p+b+q b′ for every pair of (necessarily finite) entries b,b′ in
≤ −
B and all finite entries p in P and q in Q.
Now let P′,Q′ be the matrices obtained from P and Q respectively by re-
placing each entry by δ. Thus P′,Q′ M (FT). We shall show that
n
−∞ ∈
A= P′BQ′.
Let i,j 1,...,n . By (4.1) we may choose k and l such that A =
i,j
∈ { }
P B Q P B Q for all h and m. Since A is finite it follows
i,k k,l l,j i,h h,m m,j i,j
≥
that P and Q are also finite. Thus P′ = P and Q′ = Q , giving
i,k l,j i,k i,k l,j l,j
A = P′ B Q′ . It then suffices to show that (P′BQ′) = P′ B Q′ .
i,j i,k k,l l,j i,j i,k k,l l,j
Now
n n
(P′BQ′) = P′ B Q′ ,
i,j i,h h,m m,j
Mh=1mM=1
so it will suffice to show that P′ B Q′ P′ B Q′ for all h and m.
i,k k,l l,j ≥ i,h h,m m,j
There are four cases to consider:
(a) If P ,Q FT then
i,h m,j
∈
P′ B Q′ = P B Q P B Q = P′ B Q′ .
i,h h,m m,j i,h h,m m,j i,k k,l l,j i,k k,l l,j
≤
(b) If P FT and Q = then
i,h m,j
∈ −∞
P′ B Q′ = P +B +δ
i,h h,m m,j i,h h,m
P +B +P +B +Q B P
i,h h,m i,k k,l l,j h,m i,h
≤ − −
P +B +Q = P B Q = P′ B Q′ ,
i,k k,l l,j i,k k,l l,j i,k k,l l,j
≤
by (1).
(c) If P = and Q FT then we may apply an argument dual
i,h m,j
−∞ ∈
to that in case (b), using condition (2) in place of condition (1).
(d) If P = and Q = then
i,h m,j
−∞ −∞
P′ B Q′ = δ+B +δ
i,h h,m m,j h,m
P +B +Q B +B
i,k k,l l,j h,m h,m
≤ −
P +B +Q = P B Q = P′ B Q′ ,
i,k k,l l,j i,k k,l l,j i,k k,l l,j
≤
by (3).
Thus we see that (P′BQ′) = A for all i and j.
i,j i,j
(ii) It is clear that if A B in M (T) then A B in M (T). Suppose
J n J n
≤ ≤
now that A B in M (T). Thus there exist P,Q M (T) such that
J n n
≤ ∈
A = PBQ. Let P′,Q′ be the matrices obtained from P and Q respectively
by replacing each + entry by 0. Then it is straightforward to check, by
an argument similar∞to the above, that A = P′BQ′. (cid:3)
8 GREEN’S J-ORDER AND THE RANK OF TROPICAL MATRICES
5. Characterising the -order
J
In this section we shall give an exact characterisation of the -order, and
J
hence also of the -relation, in terms of linear morphisms between column
J
spaces (or dually, row spaces). As discussed in the previous section, we
restrict our attention to the semirings FT and T, enabling us to make use
of the duality maps and the Metric Duality Theorem (Theorem 2.3 above).
Sincethe -relation inM (T)istherestrictionofthecorrespondingrelation
n
J
in M (T), this also gives a complete characterisation of .
n
J
We first recall the following result from [23].
Theorem 5.1. Let A,B M (S) for S FT,T . Then the following are
n
∈ ∈ { }
equivalent:
(i) R (A) R (B);
S S
⊆
(ii) there is a linear morphism from C (B) to C (A) taking the ith col-
S S
umn of B to the ith column of A for all i;
(iii) there is a surjective linear morphism from C (B) to C (A) taking
S S
the ith column of B to the ith column of A for all i.
We remark that Theorem 5.1 has a left-right dual, obtained by swapping
rowswithcolumnsandrowspaces withcolumnspacesthroughoutthestate-
ment. Theorem 5.1 describes a duality between embeddings of row spaces
and surjections of column spaces, which also has an algebraic manifestation:
Theorem 5.2. Let S FT,T and A,B M (S). Then the following are
n
∈ { } ∈
equivalent:
(i) C (B) surjects linearly onto C (A);
S S
(ii) R (A) embeds linearly into R (B);
S S
(iii) there exists C M (S) with A C B.
n L
∈ R ≤
Proof. Weprovefirstthat(i)implies(iii). Supposethatf : C (B) C (A)
S S
→
is a linear surjection. Let C be the matrix obtained by applying f to each
column of B. Then clearly C (C) = C (A) so C A by Theorem 3.1(iv).
S S
R
Moreover, by Theorem 5.1 and the definition of C we have R (C) R (B),
S S
⊆
so that by Theorem 3.1(i) we have C B.
L
≤
Next we show that (iii) implies (ii). Since A C, in particular A C, so
R D
Theorem 3.1(vii) tells us that there is a linear isomorphism from R (A) to
S
R (C). Also, since C B, Theorem 3.1(i) gives that R (C) is contained
S L S
≤
in R (B). Thus, the isomorphism gives a linear embedding of R (A) into
S S
R (B).
S
Finally, suppose(ii) holds,say f :R (A) R (B)isalinear embedding.
S S
→
Let A′ be obtained from A by applying f to each row of A. Then R (A) is
S
linearly isomorphic to R (A′), which is contained in R (B). By Theorem
S S
5.1,itfollowsfromthelatterthatthereisalinearsurjectionfromC (B)onto
S
C (A′). Moreover, since R (A) and R (A′) are isomorphic as S-modules,
S S S
Theorem 3.1 parts (vi) and (vii) give that C (A) and C (A′) are isomorphic
S S
as S-modules. Composing gives a linear surjection from C (B) onto C (A).
S S
(cid:3)
Weremarkthattheequivalenceofconditions(i)and(ii)inTheorem5.2is
a manifestation of a more general abstract categorical duality in residuation
GREEN’S J-ORDER AND THE RANK OF TROPICAL MATRICES 9
theory (see for example [11]). Again, the theorem has a left-right dual,
obtained by interchanging row spaces with column spaces, with , and
L R
with . Theorem 5.2 and its dual lead easily to the main result of this
L R
≤ ≤
section.
Theorem 5.3. (Linear characterisation of the -order.)
J
Let A,B M (S) for S FT,T . The following are equivalent.
n
∈ ∈ { }
(i) A B;
J
≤
(ii) there exists a convex set Y Sn such that R (A) embeds linearly
S
⊆
into Y and R (B) surjects linearly onto Y.
S
(iii) there exists a convex set Y Sn such that R (A) Y and R (B)
S S
⊆ ⊆
surjects linearly onto Y.
(iv) there exists a convex set Y Sn such that C (A) embeds linearly
S
⊆
into Y and C (B) surjects linearly onto Y.
S
(v) there exists a convex set Y Sn such that C (A) Y and C (B)
S S
⊆ ⊆
surjects linearly onto Y.
Proof. We show the equivalence of (i), (ii) and (iii), since the equivalence
of (i), (iv) and (v) is dual. Suppose that A B, say A = PBQ for some
J
≤
P,Q M (S). Thus A BQ and BQ B so that R (A) R (BQ)
n L R S S
∈ ≤ ≤ ⊆
and C (BQ) C (B) by Theorem 3.1. By the dual to Theorem 5.1, there
S S
⊆
exists a surjective linear map from R (B) onto R (BQ). Thus, setting
S S
Y = R (BQ) yields (iii).
S
That(iii) implies (ii) is trivial, so it remains only to show that (ii) implies
(i). Let Y be a convex set with the given properties. Since Y is a morphic
image of R (B), it is generated by the images of the rows of B. Thus, we
S
may suppose that Y is the row space of some matrix, say F M (S). Now
n
∈
by the dual to Theorem 5.2, there is a matrix C with F C B, and by
R
L ≤
Theorem 5.2, there is a matrix D with A D F. Thus, A B, so (i)
L J
holds. R ≤ ≤ (cid:3)
In [29] we saw that the -relation on the semigroup of 2 2 tropical
J ×
matrices was characterised by the notion of mutual isometric embedding
of projective column (dually, row) spaces. We show now that isometric
embeddingofprojectivecolumn(dually,row)spacesisanecessarycondition
of the -order, although we shall see later (Section 7 below) that it is not
J
a sufficient condition. The proof is based on the Metric Duality Theorem
from [23] (Theorem 2.3 above).
Theorem 5.4. Let A,B M (S) for S FT,T . If A B then
n J
∈ ∈ { } ≤
(i) PC (A) embeds isometrically into PC (B);
S S
(ii) PR (A) embeds isometrically into PR (B).
S S
Proof. We show that (i) holds, with (ii) being dual. Since A B there
J
≤
exist P,Q M (S) such that A = PBQ. Now A PB B so by
n R L
∈ ≤ ≤
Theorem 3.1 we have C (A) C (PB) and R (PB) R (B). It follows
S S S S
⊆ ⊆
thatPC (A) PC (PB)andPR (PB) PR (B). NowbyTheorem2.3,
S S S S
⊆ ⊆
PC (PB) is isometric to PR (PB), and PR (B) is isometric to PC (B).
S S S S
By composing inclusions and isometries in the appropriate order, we obtain
an isometric embedding of PC (A) into PC (B). (cid:3)
S S
10 GREEN’S J-ORDER AND THE RANK OF TROPICAL MATRICES
Weremarkthatthereisnoreasontobelievethattheisometricembedding
constructedintheproofofTheorem5.4isalinearmorphism. Theinclusions
are of course linear, and the isometries are “anti-isomorphisms” in the sense
of [23], but this is not sufficient to ensure that the composition is linear.
The distinction here ultimately stems from the distinction between a meet-
semilattice morphism (which by definition preserves greatest lower bounds)
and an order-morphism of meet-semilattices (which need not).
Theorem 5.2 (and also Theorem 5.3) implies that a linear embedding of
row spaces is a sufficient condition for two matrices to be related in the
-order, while Theorem 5.4 says that an isometric embedding of (projec-
J
tive) row spaces is a necessary condition for the same property. It is very
natural to ask, then, whether the former condition is necessary, or the latter
condition is sufficient. In Section 7 we shall give examples to show that, in
general, neither is the case.
6. = in the finitary case
D J
Oneof themostfundamentalstructuralquestions aboutany semigroup is
whether the relations and coincide. These relations are always equal in
D J
finite semigroups (more generally, in compact topological semigroups), but
differ in many important infinite semigroups. Semigroups in which =
D J
have an ideal structure which is considerably easier to analyse.
The full matrix semigroup M (K) of matrices over a field K is a well-
n
known example of an (infinite, provided K is infinite) semigroup in which
= . In [29] we showed that M (T) also has = , but Izhakian
2
D J D J
and Margolis [26] have recently produced examples to show that = in
M (T); it follows easily that = in M (T) for all n 3, and heDnc6e uJsing
3 n
D 6 J ≥
Proposition 4.1 and Theorem 3.2 also in M (T) for n 3. In contrast,
n
≥
in this section we shall show that the finitary tropical matrix semigroup
M (FT) satisfies = for all n.
n
D J
Our proof makes use of some topology. For convenience, we identify
PFTn−1 with Rn−1 via the correspondence
[(x ,...,x )] (x x ,x x ,...,x x ). (6.1)
1 n 1 n 2 n n−1 n
7→ − − −
Withthisidentification, theHilbertprojectivemetriconPFTn−1isLipschitz
equivalent to the standard Euclidean metric (which we will denote d ) on
E
Rn−1. Indeed, it is an easy exercise to verify that for any points x,y
Rn−1 = PFTn−1 we have ∈
d (x,y) √2d (x,y) and d (x,y) √n 1d (x,y).
H E E H
≤ ≤ −
In particular, the two metrics induce the same topology, so we may speak
without ambiguity of a sequence of points converging.
Theorem 6.1. ( = for finitary tropical matrices.)
Let A,B M (FDT). JThen A B if and only if A B.
n
∈ J D
Proof. Clearly, if A and B are -related then they are also -related. Now
D J
suppose for a contradiction that A B, but A is not -related to B. We
J D
claim first that there is an isometric (with respect to the Hilbert projective
metric) self-embedding f : PRFT(A) PRFT(A) which is not an isometry
→
(that is, which is not surjective).
